Laminate is cheap. Just replace them with new laminate counters. Laminating new laminate on top of old is doable. It just isn't easy, and certainly isn’t temporary.

Thank you. Because of the way the backsplash is installed, I can't remove the counters without majorly damaging the walls and needing a new backplash, and that's more of a job than I'm looking to do. I"m looking for a simple, cost-effective solution to cover my gross old white countertops as an intermediate-term measure!

If your substrate is still good, you can absolutely re-laminate. There are online videos of how to do this.

You cannot relaminate post formed counters with coved splashes and curved front edges. Post formed are created in a vacuum press with heat to curve them and is a thinner grade of laminate rated for curving.

To relaminate post form, both the front curved edge and the coved curved backsplash both have to be removed do that you can field create a standard flat 90° surface egde that non post form grade laminate can attach to. Those will be separate pieces, cut and trimmed with a laminate trimmer. That is quite a bit of involved work and tools for someone who doesn’t want to deal with the far easier yank and replace thr whole darn thing.

Paint it and start a piggy bank for later

https://www.youtube.com/watch?v=5I7aYosnUqI

https://www.youtube.com/watch?v=fIaepGbEGLU

Can we see how the backsplash is installed that you say is causing you an issue that would hinder replacement? Replacing the countertops is the best solution if they need to be replaced.

Jan Moyer has given good advice - paint them for now and start saving up to have the countertops and the backsplash replaced.

Many people use removable Contact paper to cover countertops for a few years. It's inexpensive. The Contact paper works best if you practice first on shelves and choose a paper with some pattern so that any tiny wrinkles or bubbles are hidden by the pattern.

Formica is pretty challenging to work with. Best to have a specific router bit and trim it after installation. There are also carbide tipped scoring tools, but those are a PITA. and work best on straights, not curves.

So the formica route is doable - but not that easy.

Go with the paint or contact paper.

Contact paper. I have thought for years that my neighbor had nice granite countertops. Nope. Contact paper. And this is a wealthy woman!

You could also use Ardex Feather Finish directly over your laminate countertops.

You can use the gray color:

You can also use the White Ardex Feather Finish:

Notice that the laminate countertops above have that extra piece of backsplash material - it was also just covered with the Ardex Feather Finish.

Before and After of the same kitchen shown the above photo:

Here is another laminate countertop that was covered with White Ardex Feather Finish:

Here is another one - looks like white Ardex Feather Finish (or maybe white with a little of the gray mixed in - or perhaps a tint was used) :

You can also purchase tints to use in cement in order to create colors other than white or gray.

I've know someone who did this in their lake house - they used a black mica powder (I believe that was the tinting agent - however, there are other types of tints specifically for cement). My lake neighbors used the regular gray and white to create veins in the tinted cement in order to create a soapstone looking countertop:

Their countertops looked very similar to the ones above (these are concrete countertops also in the photo - I just don't have a photo of my lake neighbor's countertops - they didn't install a pretty sink like in the above photo). They've had their Ardex Feather Finish "soapstone" countertops for around 5-6 years. I saw them earlier this summer - they still looked nice.

If you use a good sealer, you can prevent many of the stains that people will sometimes have with concrete countertops. Also, with the Ardex Feather Finish, you are able to fix any stains (oil or otherwise) by adding another layer over the stains. You may need to use a primer product (I can't remember what I was told) - but there are products that help new cement bond to older cement.

If your countertop has a postformed backsplash or edge, you CANNOT RELAMINTE IN THE FIELD. That technique is only accomplished inside a countertop shop and not all shops have the equipment to do it.

Your cheapest option is to purchase countertops from IKEA and replace your existing. I've had mine for 12+ years with no issues.

https://www.ikea.com/us/en/cat/pre-cut-worktops-700419/

An intermediate DIYer with a few tools could relaminate plastic laminate countertops that aren't post formed cost effectively.

And they could also cut off the nosing and backsplash to replace with hard 90° laminated pieces too.

this is simple, easy and cheap…

https://www.jamesandcolors.com/collections/countertop-contact-paper-removable-decorative-film-for-kitchens-bathrooms-and-more?srsltid=AfmBOoq4cd24B-KN5pMbS3tmOEXAc3EVLYHqEk_pq8DXF5ePqSONsxh5

Using contact paper in my classrooms, I always had to fight bubbles and wrinkles.

My temporary fix of spray appliance epoxy over cultured marble bathroom sinks has lasted 20 year with an occasional touch ups. (I would not do kitchen sink) Sand it well to start.

Rustoleum has countertop refinish kits but it is cheaper to just get their appliance epoxy. You can experiment on scrap wood or cardboard to blend bright white and biscuit for a limestone look. I use white for base coat and feather patches of biscuit on that using plastic bags or aluminium foil to sponge or rag roll for texture and blending.

Or have Miracle Method refinish them if you are not into DIY.